Distance Exceptional Graphs and the Curvature Index

TL;DR

This work introduces distance-exceptional graphs—those for which $D\vec{x}=\vec{1}$ has no solution—through the curvature index $\iota(G)$, proving $\iota(G)=0$ iff the graph is distance exceptional and developing a calculus of $\iota$ under key graph operations. It derives exact formulas for how $\iota$ behaves under Cartesian product, vertex coalescence, and graph join, enabling closure properties and constructive embeddings. The paper computes $\iota$ for diverse graph families (cycles, grids, hypercubes, trees, multipartite graphs) and shows that any rational index can be realized, while also proving that every graph can be realized as an induced (often isometric) subgraph of a distance-exceptional graph via explicit embedding procedures and an algorithmic framework. Collectively, these results provide a structured, constructive theory of distance-exceptional graphs grounded in the curvature index and graph-analytic operations, with potential applications in graph curvature and graph-embedding problems.

Abstract

A graph $G=(V,E)$ on $n$ vertices is said to be \emph{distance exceptional} if the equation $D\vec{x} = \vec{1}$ admits no solution $\vec{x}\in\mathbb{R}^{n}$, where $D\in\mathbb{R}^{n\times n}$ is the shortest path distance matrix of $G$. These graphs were first studied by Steinerberger in the context of a notion of discrete curvature (``Curvature on graphs via equilibrium measures,'' \emph{Journal of Graph Theory}, 103(3), 2023). This work has led to several open questions about distance exceptional graphs, including: What is the structure of such graphs? How can they be characterized? How rare are they? In this paper, we investigate these questions through the lens of a graph invariant we term the \emph{curvature index}. We show that a graph is distance exceptional if and only if this invariant vanishes, and we develop a calculus for this invariant under graph operations including the Cartesian product and graph join. As a result, we recover and generalize a number of known results in this area. We show that any graph $G$ can be realized as an induced subgraph of a distance exceptional graph $G'$. Moreover, in many cases, this embedding is an isometry. In turn, this leads to a number of methods for constructing distance exceptional graphs.

Figures (3)

• Figure 1: This figure contains illustrations of six distance exceptional graphs. From left to right: the two distance exceptional graphs on seven vertices, followed by two examples each on eight and nine vertices, respectively.
• Figure 2: Two illustrations of the basket graphs $\mathsf{B}_k$ with their conventional vertex enumerations. Left: $\mathsf{B}_3$. Right: $\mathsf{B}_k$. The name "basket" reflects the slightly shorter upper path serving as a "handle" above the three lower paths.
• Figure 3: This figure contains two distance exceptional graphs. (Left) A distance exceptional graph containing an induced copy of the square lattice $\mathsf{P}_5\, \Box\,\mathsf{P}_5$, formed by merging $4$ copies of $\mathsf{K}_{1, 1, 6}$ to various vertices in an arbitrary manner. (Right) A distance exceptional graph containing an induced copy of $\mathsf{Q}_4$, formed by merging $2$ copies of $\mathsf{K}_{1, 1, 6}$ to various vertices in an arbitrary manner.

Theorems & Definitions (39)

• Definition 1.2: Basket graphs
• Theorem 1.3
• Definition 1.4: Curvature index
• Theorem 1.5: See \ref{['thm:index-product']}, \ref{['thm:merging']}
• Theorem 1.6
• Theorem 1.7
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• proof